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May  2016, 10(2): 369-378. doi: 10.3934/ipi.2016004

On a transmission eigenvalue problem for a spherically stratified coated dielectric

1. 

Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, United States, United States

Received  July 2015 Published  May 2016

Suppose that the boundary of the unit ball in $R^3$ is coated with a very thin layer of a highly conductive material and the refractive index $n(x)$ inside the ball is spherically stratified. We show that in this case the set of transmission eigenvalues behave quite differently than in the previous studied case of an uncoated ball. In particular, if the index of refraction varies smoothly across the boundary of the unit ball we show that complex eigenvalues always exist and accumulate on the real axis and that the real and complex eigenvalues uniquely determine the index of refraction without any restriction on its magnitude.
Citation: David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004, 17pp. doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for spherically symmetric variable-speed wave equation, Inverse Problems, 29 (2013), 055007, 19pp. doi: 10.1088/0266-5611/29/6/065007.  Google Scholar

[3]

A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.  Google Scholar

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences Series Volume 188, Sringer, New York 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[5]

F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math Anal., 42 (2010), 2912-2921. doi: 10.1137/100793542.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I, Interscience Publishing, New York, 1953.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[8]

D. Colton and Y. J. Leung, Complex Eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008, 6pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar

[9]

D. Colton, Y. J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31 (2015), 035006, 19pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar

[10]

R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type, Bull. Amer. Math. Soc., 44 (1938), 236-240. doi: 10.1090/S0002-9904-1938-06725-0.  Google Scholar

[11]

B. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society Translation, Providence, Rhode Island, R.I., 1980.  Google Scholar

[12]

J. McLaughlin and P. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382. doi: 10.1006/jdeq.1994.1017.  Google Scholar

[13]

H. Pham and P. Stefanov, Weyl asymptotics for the transmission eigenvalues for a constant index of refraction, Inverse Prob. Imaging, 8 (2014), 795-810. doi: 10.3934/ipi.2014.8.795.  Google Scholar

[14]

W. Rundell and P. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183. doi: 10.1090/S0025-5718-1992-1106979-0.  Google Scholar

[15]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[16]

G. Wei and H. Xu, Inverse spectral analysis for the transmission eigenvalue problem, Inverse Problems, 29 (2013), 115012, 24pp. doi: 10.1088/0266-5611/29/11/115012.  Google Scholar

[17]

W. Young, Introduction to Nonharmonic Fourier Series, Academic Press, San Diego, 2001.  Google Scholar

show all references

References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004, 17pp. doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for spherically symmetric variable-speed wave equation, Inverse Problems, 29 (2013), 055007, 19pp. doi: 10.1088/0266-5611/29/6/065007.  Google Scholar

[3]

A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.  Google Scholar

[4]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences Series Volume 188, Sringer, New York 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[5]

F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math Anal., 42 (2010), 2912-2921. doi: 10.1137/100793542.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I, Interscience Publishing, New York, 1953.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[8]

D. Colton and Y. J. Leung, Complex Eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008, 6pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar

[9]

D. Colton, Y. J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31 (2015), 035006, 19pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar

[10]

R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type, Bull. Amer. Math. Soc., 44 (1938), 236-240. doi: 10.1090/S0002-9904-1938-06725-0.  Google Scholar

[11]

B. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society Translation, Providence, Rhode Island, R.I., 1980.  Google Scholar

[12]

J. McLaughlin and P. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382. doi: 10.1006/jdeq.1994.1017.  Google Scholar

[13]

H. Pham and P. Stefanov, Weyl asymptotics for the transmission eigenvalues for a constant index of refraction, Inverse Prob. Imaging, 8 (2014), 795-810. doi: 10.3934/ipi.2014.8.795.  Google Scholar

[14]

W. Rundell and P. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 58 (1992), 161-183. doi: 10.1090/S0025-5718-1992-1106979-0.  Google Scholar

[15]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[16]

G. Wei and H. Xu, Inverse spectral analysis for the transmission eigenvalue problem, Inverse Problems, 29 (2013), 115012, 24pp. doi: 10.1088/0266-5611/29/11/115012.  Google Scholar

[17]

W. Young, Introduction to Nonharmonic Fourier Series, Academic Press, San Diego, 2001.  Google Scholar

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